Instantaneous Speed, Velocity and Acceleration

Traveling in vehicles has become a regular part of our lives

Let us suppose we are driving a car from point A to B

The car covers this distance of 50km in time 1 hr

When an object covers a distance in a given interval of time, we can define Average Speed of the object

Mathematically, $A v e r a g e s p e e d = \frac{T o t a l d i s t a n c e c o v e r e d}{T o t a l t i m e t a k e n}$

Therefore, the Average Speed of the Car $= \frac{5 0 k m}{1 h r} = 50 k m / h r$

The Average Speed does not indicate the entire kinematics of the car's motion

The Average Speed does not mean that the car is traveling at the same speed all the time.

There might be times when the car has to speed up (like on a highway), or it has slowed down (on a signal).

Let us consider a point C between A and B where there is a traffic signal

So, at point C, the car has to slow down from a certain finite speed to 0

Even though our Average Speed is 50km/hr, but at point C, Speed of the car is 0km/hr

Therefore, the Speed of the car at point C is known as its Instantaneous Speed

The Instantaneous Speed is the speed of an object at a certain instant of time

Using a similar approach, let us try to understand the concept of velocity

We know, Velocity is Speed with a specified direction

For example, three Trains X,Y and Z are moving with 100km/hr but traveling to different cities

Here, the Speed of the trains are the same but their direction is different

Hence, the velocities of the trains are different.

So on similar lines, we can define the Instantaneous Velocity of an object to be the velocity of an object at a certain instant of time

When the velocity of an object changes with time, we get Acceleration

Considering the previous example, when we had to stop our car at point C, we had to apply brakes

Because of this brakes applied, the velocity changed from some finite value to 0

So, when there is a change of Velocity of an object with time, we say the object to be "Accelerated"

Mathematically, $A c c e l e r a t i o n = \frac{C h a n g e i n V e l o c i t y}{T i m e}$

Instantaneous Acceleration is thus defined as the acceleration of an object at a specific instant of time

Let us look into the concept of Instantaneous quantities with the help of an example

Firstly, we give a more formal mathematical approach of defining Instantaneous quantities like Instantaneous Velocity, Instantaneous Acceleration etc

Instantaneous Velocity: $lim_{Δ t \to 0} (\frac{Δ x}{Δ t}$ ) = $\frac{d x}{d t}$

where $Δ x$ is the change in displacement vector and $Δ t$ is the change in time

It is the velocity of the object, calculated in the shortest instant of time possible (calculated as the time interval $Δ t$ tends to zero).

Mathematically, $lim_{Δ t \to 0} \frac{Δ S}{Δ t} = \frac{d S}{d t}$

Let's solve a problem on instantaneous velocity for better understanding

Q: A bullet fired in space is traveling in a straight line and its equation of motion is $S (t) = 4 t + 6 t^{2}$

If it travels for 15 seconds before impact, find the instantaneous velocity and acceleration at the 10th second

Solution: We know the equation of motion: $S (t) = 4 t + 6 t^{2}$ $\frac{d S}{d t} = \frac{d ( 4 t + 6 t ^{2} )}{d t} = 4 + 12 t$

Therefore, $V_{I n s t}$ at (t =10) = 4 + (12 x 10) = 124 m/s. The Instantaneous Velocity of the bullet is 124 m/s

Instantaneous Acceleration is defined as: $lim_{Δ t \to 0} \frac{Δ V}{Δ T}$ , where $Δ V$ is the change in Velocity vector

Mathematically, $lim_{Δ t \to 0} \frac{Δ V}{Δ T} = \frac{d V}{d T} = \frac{d ^{2} S}{d ^{2} T}$ In the problem, $\frac{d ^{2} S}{d ^{2} T} = 12$

Hence, Instantaneous Acceleration of the bullet is $12 m / s^{2}$

Revision

Average Speed is defined as the ratio of the total distance covered to the total time taken. It does not indicate constant velocity of travel

Velocity of a particle/object is a vector with a specified direction. It's magnitude is the speed of the particle/object. Hence, Velocity is nothing but speed with a direction

Instantaneous quantities are defined at an instant of time. Examples: Instantaneous Velocity of an object is defined as it's Velocity at an instant of time

The End